Rodgers was unhappy that Dowd awarded two second-half penalties, the first of which also saw Daniel Agger red-carded for a professional foul on Albion striker Shane Long.
Although Long's resulting penalty was saved by Jose Reina, Peter Odemwingie tucked away the second spot-kick, awarded for Martin Skrtel's foul on Long, while Albion also found the target through Zoltan Gera and Romelu Lukaku as Steve Clarke enjoyed a dream start as Baggies boss.
Rodgers, however, felt the margin of victory did not reflect his side's efforts.
He said: "I thought the scoreline was harsh. Congratulations to West Brom and Steve Clarke getting his first win as a manager.
"I've no complaints about the first goal, a fantastic strike from Gera, showing wonderful technique.
"But in the second half, although I'm not going to go on about the referee, I thought the two penalty decisions were very harsh.
"I don't think either was a penalty and Martin Skrtel doesn't even connect for the second one.
"With the first one, Long's pace has got him there. There is slight contact but I know Long well, how big and strong he is, and it was very harsh.
"With 10 men and then chasing the game, it becomes difficult."
Rodgers admitted Liverpool may consider appealing Agger's red card, saying: "It is something we will look at. It was very harsh."
The former Swansea boss admits his Liverpool side remain a work in progress but still saw some encouraging signs in the opening 45 minutes.
He said: "We showed some signs of our quality but the penalties and the sending-off kills you, especially a new team coming together.
"But I've got to give big credit to the players. They ran to a standstill today and I can't complain.
"It is still a work in progress. It is very much a marathon we are in. This is the first game.
"We've got to win games. I've heard lots about patience and every manager wants that. But nowadays you don't get that. We've got to win games. It was a bad day in the office."
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Reds striker Luis Suarez missed several chances and was booked on the stroke of half-time, but Rodgers said: "When you are a winner with so much passion, a game like today hurts.
"He is a wonderful player, a good man, world class and a winner and when it goes against you, it can be difficult."
To add to Rodgers' woes, substitute Joe Cole limped off with a recurrence of a hamstring problem.
Meanwhile, Baggies boss Clarke insisted he gained no extra satisfaction from defeating Liverpool.
Clarke, who was number two to Kenny Dalglish at Anfield last season, said: "The manner of departure from Liverpool is not an issue. That's what happens in football these days.
"It doesn't give me any extra satisfaction winning this game. If I was sitting here and we had played someone else and won, I would be just as happy.
"I've got no axe to grind with Liverpool."
Clarke was unsurprised by the quality of his side's performance.
He said: "I think I've been in the game long enough to know you enjoy these moments but you know there are different Saturdays to come.
"But the performance today showed these players are comfortable in the Premier League.
"They've been there for two seasons, they've finished mid-table both times, they know they can handle themselves at this level.
"I wasn't worried about the game. I knew it was a good game for us to start with, that there was more pressure on Liverpool than us and that proved to be the case.
"But I was really pleased with the way we played."